In Serge Lang's Algebra on page 772 in the middle there is an expression of this form
I don't understand what $Z[A(C)]$ is supposed to mean. Thanks.
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The $Z$ denotes the integers, and $[A(C)]$ the set of isomorphism classes of objects in $A(C)$. Then $Z[A(C)]$ is just the free abelian group with generators the elements of $[A(C)]$.